With the clarification made that we are talking about composition length, and that the socle should have finite length, the proposed statement is false.
A module is called finitely cogenerated if it has a finitely generated essential socle. (Here the "finitely generated" may be swapped for "finite composition length", since the two notions are identical for a semisimple module.)
So, the question amounts to asking "Show a module is Artinian iff it is finitely cogenerated." From Mariano's answer, we know that Artinian modules are finitely cogenerated, but the converse is false. A correct statement would be that "a module $M$ is Artinian iff $M/N$ is finitely cogenerated for every submodule $N$ of $M$."
There exists a non-Artinian commutative ring $R$ such that $R_R$ is a finitely cogenerated module. The first example I located is due to Osofsky and can be found in Lam's Lectures on Modules and Rings at the top of page 514.
If you request, I can duplicate it here.